Alethograph

Measure Theory - Bogachev

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Textbook

Author: Vladimir I. Bogachev | Publisher: Springer, 2007 | Volumes: 2

Overview

A comprehensive two-volume treatise on modern measure theory covering the classical theory (Volume 1: Chapters 1-5) and more advanced topics including measures on topological spaces, weak convergence, transformations, and conditional measures (Volume 2: Chapters 6-10). The standard graduate-level reference for measure theory, with extensive supplements and over 850 exercises.

Topics Studied

measure convergence (studied 21-03-2026)

Chapters read: Ch. 7.9-7.11 (pp. 108-117), Ch. 8.1-8.9 (pp. 175-213)

Key Definitions

  • Definition 8.1.1 (p. 175): weak convergence of measures — a net {mu_alpha} in M_sigma(X) converges weakly to mu if integral f d(mu_alpha) → integral f d(mu) for all f in C_b(X). Notation: mu_alpha ⇒ mu
  • Definition 8.1.2 (p. 175): Weak topology — the topology sigma(M_sigma(X), C_b(X)) on the space of Baire measures; this is the weak-* topology of functional analysis
  • Definition 7.9.2 (p. 109): Three classes of functionals on C_b(X): sigma-smooth (L(f_n) → 0 for f_n decreasing to 0), tau-smooth (same for nets), and tight (f_alpha → 0 uniformly on compacts implies L(f_alpha) → 0). These correspond to Baire, tau-additive, and Radon measures
  • Definition 8.6.1 (p. 202): Uniform tightness — a family M of Radon measures is uniformly tight if for every eps > 0 there exists compact K_eps with |mu|(X \ K_eps) < eps for all mu in M

Key Theorems

  • Theorem 7.9.1 (p. 108): Alexandroff representation — bounded linear functionals on C_b(X) correspond bijectively to additive regular set functions on the Baire algebra, with norm preservation. Positive functionals correspond to nonneg set functions.
  • Theorem 7.10.1 (p. 111): Baire measures correspond bijectively to sigma-smooth functionals on C_b(X)
  • Theorem 7.10.4 (p. 111): Riesz representation on compact spaces — every continuous linear functional on C(K) is represented by a unique Radon measure
  • Theorem 7.10.6 (p. 112): On completely regular spaces, Radon measures correspond to tight functionals on C_b(X)
  • Theorem 7.11.1 (p. 114): On locally compact X, a set function on compact sets satisfying subadditivity and additivity on disjoint compacts extends uniquely to an outer regular Borel measure with mu(U) = sup{mu(K): K compact, K in U}
  • Theorem 7.11.3 (p. 116): Riesz representation on locally compact spaces — every nonneg linear functional on C_0(X) is represented by a Borel measure on B(X), which can be chosen Radon and inner compact regular
  • Proposition 8.1.7 (p. 177): Banach-Steinhaus for measures — if sup_{mu in M} integral f d(mu) < infinity for all f in C_b(X), then sup ||mu|| < infinity. Every weakly convergent sequence of Baire measures is bounded in variation norm.
  • Proposition 8.1.8 (p. 177): [[weak convergence of measures|Weak convergence of signed measures on [a,b]]] — a sequence of signed measures mu_n converges weakly precisely when sup_n ||mu_n|| < infinity and every subsequence of distribution functions F_{mu_n} has a further subsequence converging to F_mu at all points except an at most countable set. For nonneg measures, F_{mu_n} → F_mu at all continuity points.
  • Theorem 8.2.3 (p. 184): portmanteau theorem for probability measures on topological spaces — weak convergence iff lim sup mu_alpha(F) ⇐ mu(F) for functionally closed F iff lim inf mu_alpha(U) >= mu(U) for functionally open U. For nonneg non-probability measures, need lim mu_alpha(X) = mu(X) additionally.
  • Corollary 8.2.4 (p. 184): On metrizable (or perfectly normal) spaces, functionally closed/open can be replaced with closed/open in the portmanteau conditions. Same holds if measures are tau-additive (e.g. Radon).
  • Theorem 8.2.7 (p. 186): Weak convergence of probability measures iff lim mu_alpha(E) = mu(E) for every E in Ba(X) with the property that there exist functionally open W and functionally closed F with W in E in F and mu(F \ W) = 0
  • Corollary 8.2.10 (p. 187): portmanteau theorem on metrizable spaces — for probability measures, weak convergence iff lim sup mu_alpha(F) ⇐ mu(F) for closed F iff lim inf mu_alpha(U) >= mu(U) for open U iff lim mu_alpha(E) = mu(E) for all mu-continuity sets E
  • Corollary 8.2.11 (p. 187): On R, probability measures converge weakly iff distribution functions converge at continuity points
  • Theorem 8.3.2 (p. 193): Prokhorov metric — the Levy-Prohorov metric metrises the weak topology on nonneg tau-additive measures on any metric space. On separable spaces, d_0(mu,nu) = ||mu - nu||_0 also metrises. The weak topology on P_sigma(X) is NOT metrizable if P_sigma(X) != P_tau(X).
  • Theorem 8.4.7 (p. 197): Varadarajan’s theorem for signed measures — if mu_alpha converges weakly to mu, then lim inf |mu_alpha|(U) >= |mu|(U) for every functionally open U. Moreover, |mu_alpha| converges weakly to |mu| precisely when |mu_alpha|(X) → |mu|(X).
  • Corollary 8.4.8 (p. 198): Mass preservation for H-J parts — if mu_alpha ⇒ mu weakly and |mu_alpha|(X) → |mu|(X), then mu_alpha^+ ⇒ mu^+ and mu_alpha^- ⇒ mu^- weakly
  • Theorem 8.6.2 (p. 202): Prohorov’s theorem — on a complete separable metric space, a family M of Borel measures has the property that every sequence in M contains a weakly convergent subsequence iff M is uniformly tight and uniformly bounded in variation
  • Theorem 8.6.4 (p. 204): Le Cam’s strengthening — for nonneg Radon measures on any metric space, weak convergence implies uniform tightness (completeness not needed)
  • Theorem 8.9.4 (p. 213): M_tau^+(X) with weak topology is metrizable iff X is metrizable; a complete metric exists iff X has one

Key Examples

  • Example 8.1.4 (p. 176): nu_n with densities np(nt) converge weakly to delta_0 — illustrates mass concentration
  • Example 8.4.5 (p. 197): (i) mu_n = sin(nx)dx on [0,2pi] converges weakly to 0 but |mu_n| has no weak limit; (ii) delta_0 - delta_{1/n} → 0 weakly but |delta_0 - delta_{1/n}| → 2*delta_0. Shows Hahn-Jordan decomposition is NOT continuous under weak convergence.
  • Example 8.4.6 (p. 197): Le Cam’s example on a compact metric space X where mu_n ⇒ 0 but |mu_n| ⇒ mu which is tau-additive but NOT Radon
  • Example 8.6.9 (p. 207): On a countable nonmetrizable space, a weakly convergent sequence of probability measures may not be uniformly tight (showing completeness/metrizability is essential for Prohorov)

Key Observations for Herdegen-Liang-Shelley Paper

  • Proposition 8.1.8 (p. 177) is a precursor to basic convergence: it characterises weak convergence of signed measures on [a,b] via subsequential distribution function convergence at all but countably many points + uniform boundedness in variation. This is essentially Stanek’s Theorem 3.12 restricted to compact intervals, predating both Khartov and Stanek.
  • Bogachev works on general topological spaces (not just metrisable), using Baire sigma-algebra Ba(X) and functionally closed/open sets. The portmanteau theorem (Theorem 8.2.3) requires functionally closed/open sets on general spaces, reducing to closed/open on metrizable spaces (Corollary 8.2.4). Your paper works on metrisable spaces, so this distinction does not arise, but noting it provides context.
  • Theorem 8.4.7 (Varadarajan) and Corollary 8.4.8 give the general topological space version of your Table 1 results. Your contribution is the vague convergence side (which Bogachev does not treat — he only considers weak convergence here).
  • The remark on p. 182 that the weak topology on signed measures is not metrizable (Exercise 8.10.72) is confirmed independently by Stanek (Theorem 3.1). Bogachev’s exercise predates Stanek by ~17 years.
  • Local compactness matters precisely because C_0(X) separates points only when X is locally compact (Section 7.11, p. 114). On non-locally-compact spaces, C_0(X) = {0}, making vague convergence (testing against C_c or C_0) trivially satisfied. This is why your paper’s definition uses C_c(Omega) on metrisable spaces rather than C_0.
  • The decomposition of functionals into sigma-smooth / tau-smooth / tight (Definition 7.9.2, Theorems 7.9.3-7.9.5) provides a systematic framework for understanding which functionals correspond to which types of measures. This could augment your paper’s discussion of the Riesz-Markov-Kakutani theorem (your Theorem 1.2).

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